In computational fluid dynamics (CFD) and nonlinear fluid-structure interaction (FSI), embedded boundary methods (EBMs) are Eulerian approaches that operate on non-body-fitted fluid meshes, embedding discrete representations of obstacle surfaces. They offer exceptional robustness for problems involving large motions, deformations, shape modifications, or surface topology changes, while enabling highly automated discretization of complex geometries – approaching a meshing-free workflow. These capabilities make EBMs particularly attractive for multidisciplinary design, analysis, and optimization (MDAO). Despite these advantages, conventional EBMs often suffer from discrete events that induce spurious oscillations and compromise differentiability near embedded boundaries, restricting their applicability in gradient-based MDAO. Furthermore, high-dimensional EBMs remain computationally expensive for parametric or high Reynolds number flows, and projection-based model order reduction (PMOR) – well-established for body-fitted CFD – has seen limited application to non-body-fitted CFD methods due to dynamic domain partitioning and associated challenges in snapshot collection and reduced-basis construction. This lecture presents FIVER++, a novel framework for constructing differentiable, discrete-event-free EBMs. FIVER++ combines a smoothness-indicator nodal function with a moving least-squares procedure to suppress spurious oscillations in integral quantities computed on embedded interfaces. To accelerate high-fidelity computations for parametric and high Reynolds number flows, this differentiable EBM is coupled with a PMOR approach, referred to here as PMOR-EBM. The resulting FIVER++/PMOR-EBM framework leverages the physics-based machine learning method ECSW (energy-conserving sampling and weighting) for hyperreduction of repeated nonlinear projections and employs a piecewise-affine low-dimensional approximation to mitigate the Kolmogorov n-width barrier of transport-dominated flows. The FIVER++/PMOR-EBM framework is demonstrated on challenging FSI and MDAO problems, including transonic limit-cycle oscillations of complete fighter-jet configurations and shape-parametric steady-state and unsteady turbulent flow studies, showcasing its potential for rapid, high-fidelity predictions in complex engineering design scenarios.